Evaluate ∫ from 1 to 2 of 3/(x^2 + 3x) dx Step by Step

Step by step

The question

Question: find the integral from 1 to 2 of 3, all over x² + 3x.

∫ from 1 to 2 of 3x² + 3x dx =

1 Factor the bottom

First factor the bottom.

We need two numbers that multiply to 0 and add to 3: 3 and 0.

x² + 3x = (x + 3)x

3 × 0 = 0, 3 + 0 = 3

2 Split the fraction

Two simple fractions can add up to this one: A over (x + 3), plus B over x.

We find the numbers A and B.

3(x + 3)x = A(x + 3) + Bx

Careful!

ln of the bottom works only when the top is the bottom's derivative.

Here it is not, so we split.

Careful: not ∫ 3x² + 3x dx = ln|x² + 3x| ✗

3 Clear the bottoms

Multiply both sides by (x + 3)x.

On the left the bottom goes.

On the right each fraction keeps the other bracket.

3 = Ax + B(x + 3)

4 Find A and B

Put in x = −3: it makes (x + 3) zero, so the B part is gone.

x = −3: 3 = A(−3) + B(0)

Divide by −3: A = −1.

A = 3−3 = −1

Put in x = 0: it makes x zero, so the A part is gone.

x = 0: 3 = B(3) + A(0)

Divide by 3: B = 1.

B = 33 = 1

5 Integrate

Now the integral is two easy pieces.

∫ from 1 to 2 of 3(x + 3)x dx = ∫ from 1 to 2 of (−1(x + 3) + 1x) dx

One over x minus a number integrates to ln of it, like 1 over x.

∫ 1x − p dx = ln|x − p| + C

No + C: top minus bottom.

= [ −ln|x + 3| + ln|x| ] from 1 to 2

6 Top minus bottom

Put in 2, then take away the same with 1, for each piece.

= −(ln 5 − ln 4) + (ln 2 − ln 1)

Two logs taken away make one log of a fraction.

ln a − ln b = ln ab

The two numbers in front are the same size, 1.

Take it out: one log minus the other is the log of their fraction.

= −ln 54 + ln 2

So the integral is ln of 8/5, about 0.470.

= ln 85 ≈ 0.470

7 Check

Let's check: add the two fractions back over one bottom, (x + 3)x.

−1(x + 3) + 1x = −x + (x + 3)(x + 3)x

Open the brackets and add: the top is 3 again.

Right!

= 3(x + 3)x

Let's look back.

First: factor the bottom.

Then: split the fraction.

Then: clear the bottoms.

Then: find A and B.

Then: integrate.

Then: top minus bottom.

Last: check.

Answer